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In mathematics, one often makes the remark that what is being talked about is a perfect idealized object. "Our planet is a sphere, but it's not really a perfect mathematical sphere (that is perfectly smooth, etc)."

What would one say, then, about natural numbers? In what way are they perfect?

My understanding is that there might be, e.g., two somethings in front of you, but these two somethings are never really the same somethings (e.g., because they are in two different places in space at the same time, so they are not really the same object). And because arguably, the number 2 does not exist inside of space and time, 2 represents "perfect twoness", so to speak.

Is my understanding correct? If not, what would a correct answer be?

In what sense are natural numbers perfect for the mathematical platonist?

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    Accoriding to SEP perfect is for their independence of humans plato.stanford.edu/entries/platonism-mathematics. I’d also say their lacking of physical properties makes them perfect, going back to Plato like nwr said. They don’t have any chaotic features of the physical world. (But we really shouldn’t say they are perfectly smooth in the physical sense either as they have no physical properties). In this sense only abstract objects are perfect, while electrons (physical) can only be perfectly independent.
    – J Kusin
    Jul 9, 2023 at 21:35
  • Are you implying that two as observed in reality is necessarily not a "perfect (idealised) two"? I'm not sure there has to be a distinction like this. If you count two items of some kind, why can't the resulting number two be as perfect as it gets? (But then I'm not a Platonist.) Jul 9, 2023 at 22:00
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    @MauroALLEGRANZA I will err on the side of Plato with this one.
    – user42828
    Jul 10, 2023 at 14:39
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    @Ruse, yeah, I know what perfect numbers are...
    – user42828
    Jul 10, 2023 at 17:50
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    I come from the mathematics world and am not well versed in philosophy, in particular I don't know what platonist means, but I will add what I think is the most fundamental property of the natural numbers, which is that they are (up to isomorphism) the minimally inductive set. Which loosely translates to them being the smallest infinite set
    – Carlyle
    Jul 13, 2023 at 7:50

2 Answers 2

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I think Plato uses the idea of "perfection" of forms multiple ways. I think one of them is the way you described above. ie: in the macroscopic world anyway, you're right. Within our sense experience do we ever really see two things? (do we even see distinct objects). We "apply" the concept of two based on some kind of categorization of objects. And this categorization isn't hard and fast. We know in the microscopic world there are identical objects like electrons. But these are outside of sensation except in an indirect sense.

Generally we apply the purely abstract concept of two onto a particular situation according to our particular interest at the time. For example I say there are two objects on the table, ignoring all the dust particles that may also be there.

There is an inherent "fuzziness" to our sense-experience onto which we apply these "perfect" concepts (eg: square, triangle, circle, three). Not quite sure how we do this. Some concepts at least mathematical ones don't seem to have this fuzziness.

But I think the main way Plato uses the word perfect is... timeless and unchanging... whereas two cups or two animals will eventually go out of existence, the number two itself will always remain itself, unchanging.

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The natural numbers (positive integers excluding zero) are for counting objects, like for example goats. Your goat herd might contain goats of various sizes and colors but the idealized ("perfect") herd count will always be a positive integer and will always include all the goats.

That said, no number in the set of reals for example enjoys a more "perfect" status than any other number in that set, and so while the idea of a "perfect number" might have had meaning to a Greek philosopher musing on the size of a herd of goats 3000 years ago, that idea is mathematically meaningless today.

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